Curl

Mathematics

Curl

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The curl is a way of expressing a certain type of derivative of a vector field. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.

The curl is written as though it were the cross product of the special symbol "$\nabla$" (which is commonly called "del" or "nabla"), with the given vector field, like this: $\nabla \times {\vec {V}}$. This is usually pronounced "curl V" or "del cross V".

In ordinary Cartesian coordinates, the curl is calculated as:

$\nabla \times {\vec {V}}=\left({\frac {\partial V_{z}}{\partial y}}-{\frac {\partial V_{y}}{\partial z}},\ {\frac {\partial V_{x}}{\partial z}}-{\frac {\partial V_{z}}{\partial x}},\ {\frac {\partial V_{y}}{\partial x}}-{\frac {\partial V_{x}}{\partial y}}\right)$

or, using a somewhat fictitious notation like the determinant notation for cross product,

$\nabla \times {\vec {V}}={\begin{vmatrix}{\hat {x}}&{\hat {y}}&{\hat {z}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\V_{x}&V_{y}&V_{z}\end{vmatrix}}$

where ${\hat {x}}$, ${\hat {y}}$, and ${\hat {z}}$ are the unit basis vectors.

If one thinks of $\nabla$ as being a fictional vector field with components $\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$, one can sort of see that the cross product notation makes sense. This is also useful for remembering how to calculate a curl.

The curl is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The curl operation has an intrinsic "handedness" to it. Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".

The curl is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in Maxwell's Equations.

Intuitively, the curl measures the degree to which the vector field rotates around a given point. If you were to measure the curl of the vector field of wind speed in the vicinity of a meteorological low pressure area, then, keeping in mind that the Coriolis force makes the wind move in a counterclockwise vortex in the Northern hemisphere, it would be a vector pointing upward. (The reason for the Coriolis force is not important here, we're just talking about the observation that the air moves in a counterclockwise vortex.) The way to see that the curl points upward is to visualize a giant right hand with the fingers curled counterclockwise—the thumb would point upward.

Vector fields with a curl of zero are called irrotational.